91Ó°ÊÓ

When solving equations, one of the crucial steps is isolating the terms. This means you want to get the term with the variable you are solving for by itself on one side of the equation.
In the given problem, we start with the equation: \[ x + \sqrt{x} = 20 \]The goal is to isolate the square root term. By subtracting \(x\) from both sides, we achieve this: \[ \sqrt{x} = 20 - x \]
Isolating terms simplifies the equation and makes the next steps more manageable.

The quadratic formula is a powerful tool used to solve quadratic equations of the form \(ax^2 + bx + c = 0\). The general form of the quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]In the problem, after isolating and squaring the terms, we get a quadratic equation: \[ x^2 - 41x + 400 = 0 \]Here, \(a = 1\), \(b = -41\), and \(c = 400\). We plug these values into the quadratic formula: \[ x = \frac{41 \pm \sqrt{41^2 - 4 \times 1 \times 400}}{2 \times 1} \]The discriminant (the part under the square root) will help us determine if there are real solutions: \[ 41^2 - 4 \times 1 \times 400 = 1681 - 1600 = 81 \]Since the discriminant is positive, we have two real solutions. These solutions are: \[ x = \frac{41 + 9}{2} = 25 \] and \[ x = \frac{41 - 9}{2} = 16 \]

Once we have potential solutions, it's essential to verify if they are real and valid.
We found two potential solutions from the quadratic formula: \(x = 25\) and \(x = 16\).
To verify, we substitute these back into the original equation to check if they work: \[ 25 + \sqrt{25} = 25 + 5 = 30 \quad \text{(Not a valid solution)} \] \[ 16 + \sqrt{16} = 16 + 4 = 20 \quad \text{(Valid solution)} \] Hence, the only **real solution** to the original equation \(x + \sqrt{x} = 20\) is \(x = 16\).